Chapter 10

Arbitrage Pricing Theory
and
Multifactor Models of Risk and Return
GVHD: TS. Trần Thị Hải Lý

Nhóm 5
Lâm Bá Du
Huỳnh Thái Huy
Phan Tuyết Trinh
Trần Thị Ngọc Hạnh
Tô Thị Phương Thảo
Nguyễn Hoàng Minh Huy


OUTLINE

Multifactor Models: An Overview

Arbitrage Pricing Theory

The APT, the CAPM, and the Index Model

A Multifactor APT

The Fama-French (FF) Three-Factor Model


10.1 Multifactor Models: An Overview



Ri nonsystematic components of returns, the e , are assumed to be uncorrelated across
The
i
stocks and with the factor F.


10.1 Multifactor Models: An Overview

SINGLE-FACTOR MODEL

Ri = E(Ri) + βiF + ei (1)
F: the deviation of the common factor from its expected value.
βi: the sensitivity of firm i to that factor.
ei: the firm-specific disturbance.
The actual excess return on firm i will equal its initially expected value plus a (zero
expected value) random amount attributable to unanticipated economywide events, plus
another (zero expected value) random amount attributable to firm-specific events


10.1 Multifactor Models: An Overview
SINGLE-FACTOR MODEL

Ex:
Ri = E(Ri) + βiF + ei (1)
Suppose F is taken to be news about the state of the business cycle.

•
•



two-factor model

macroeconomic sources of risk are measured by unanticipated growth in GDP and
unexpected changes in interest rates IR
The return on any stock will respond both to sources of macro risk and to its own
firm-specific influences. Then:
Ri = E(Ri) + βiGDPGDP + βiIRIR +ei (2)


10.1 Multifactor Models: An Overview
MULTIFACTOR MODELS
two-factor model
two-factor model

Ri = E(Ri) + βiGDPGDP + βiIRIR +ei (2)

•
•

Both macro factors have zero expectation

•
•

An increase in interest rates is bad news for most firms

βiGDP and βiIR measure the sensitivity of share returns to that factor
loadings or factor betas.
ei reflects firmspecific influences.

Ri = E(Ri) + βiGDPGDP + βiIRIR +ei (2)
Suppose the result of Northeast Airlines estimation by using multifactor
models is

•
•
•

R = .133 + 1.2(GDP) - .3(IR) + e
E(R) for Northeast is 13.3%
With every percentage point increase in GDP beyond current
expectations, the return on Northeast shares increases on average by
1.2%,
With every unanticipated percentage point that interest rates
increases, Northeast’s shares fall on average by .3%.


10.1 Multifactor Models: An Overview
MULTIFACTOR MODELS
two-factor model

•
•

Ri = E(Ri) + βiGDPGDP + βiIRIR +ei (2)
where E ( R) comes from? What determines a security’s expected
excess rate of return.
This is where we need a theoretical model of equilibrium security
returns
arbitrage pricing theory can help determine the

Arbitrage
Pricing Theory (APT)

Ross’s APT relies on three key propositions:

(1)
(2)
(3)

security returns can be described by a factor model;
There are sufficient securities to diversify away idiosyncratic risk;
Well-functioning security markets do not allow for the persistence of arbitrage
opportunities.


10.2 Arbitrage Pricing Theory (APT)
Single- Factor APT Model

••

We begin with a simple version of Ross’s model, which assumes that only one systematic

 

factor affects security returns.
(10.4)
(10.5)


10.2 Arbitrage Pricing TheoryArbitrage


10.2 Arbitrage Pricing TheoryArbitrage
(APT)Pricing Theory
•
•

The Law of One Price states that
if two assets are equivalent in all economically relevant respects, then they should have
the same market price.

•
•

The Law of One Price is enforced by arbitrageurs:
If they observe a violation of the law, they will engage in arbitrage activity simultaneously
buying the asset where it is cheap and selling where it is expensive. In the process, they
will bid up the price where it is low and force it down where it is high until the arbitrage
opportunity is eliminated.


10.2 Arbitrage Pricing TheoryArbitrage
(APT)Pricing Theory
They will engage in arbitrage activity
simultaneously buying the asset
where it is cheap and selling where
it is expensive. In the process, they
will bid up the price where it is low
and force it down where it is high
until the arbitrage opportunity is
eliminated


We can divide the variance of this portfolio into systematic and nonsystematic sources:

 

Where:

•
•

is the variance of the factor F
is the nonsystematic risk of the portfolio, with,


10.2 Arbitrage Pricing TheorySingle(APT)
Factor APT Model

•

If the portfolio were equally weighted, =1/ n, then the nonsystematic variance would be

 


10.2 Arbitrage Pricing TheorySingle(APT)
Factor APT Model
••

Because the expected value of for any well-diversified portfolio is zero, and its variance